UNIT 1.5 ADDING AND SUBTRACTINGUNIT 1.5 ADDING AND SUBTRACTING

REAL NUMBERSREAL NUMBERS

Warm UpSimplify.

1.

3 2. –4

Write an improper fraction to represent eachmixed number.

3. 423

143

4. 767

557

Write a mixed number to represent each improper fraction.

5. 125 2

25

6. 249

223

|–3| –|4|

Add real numbers.

Subtract real numbers.

Objectives

Vocabularyabsolute valueoppositesadditive inverse

All the numbers on a number line are called realnumbers. You can use a number line to modeladdition and subtraction of real numbers.

Addition

To model addition of a positive number, move right. To model addition of a negative number move left.

Subtraction

To model subtraction of a positive number, move left. To model subtraction of a negative number move right.

Example 1A: Adding and Subtracting Numberson a Number line

Add or subtract using a number line.

Start at 0. Move left to –4.

11 10 9 8 7 6 5 4 3 2 1 0

+ (–7)

–4+ (–7) = –11

To add –7, move left 7 units.

–4

–4 + (–7)

Example 1B: Adding and Subtracting Numberson a Number line

Add or subtract using a number line.

Start at 0. Move right to 3.

To subtract –6, move right 6 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

+ 3

3 – (–6) = 9

3 – (–6)

–6

Add or subtract using a number line.

–3 + 7

Check It Out! Example 1a

Start at 0. Move left to –3.

To add 7, move right 7 units.

-3 -2 -1 0 1 2 3 4 5 6 7 8 9

–3

+7

–3 + 7 = 4

Check It Out! Example 1b

Add or subtract using a number line.

–3 – 7 Start at 0. Move left to –3.

To subtract 7 move left 7 units.

–3–7

11 10 9 8 7 6 5 4 3 2 1 0

–3 – 7 = –10

Check It Out! Example 1c

Add or subtract using a number line.

–5 – (–6.5) Start at 0. Move left to –5.To subtract negative 6.5 move right 6.5 units.

8 7 6 5 4 3 2 1 0

–5

–5 – (–6.5) = 1.5

1 2

– (–6.5)

The absolute value of a number is the distance from zero on a number line. The absolute value of 5 is written as |5|.

5 units 5 units

210123456 6543- - - - - -

|5| = 5|–5| = 5

Example 2A: Adding Real Numbers

Add.

Use the sign of the number with the greater absolute value.

The sum is negative.

When the signs of numbers are different, find the difference of the

absolute values:

Example 2B: Adding Real Numbers

Add.

y + (–2) for y = –6

y + (–2) = (–6) + (–2)

(–6) + (–2)

First substitute –6 for y.

When the signs are the same, find the sum of the absolute values: 6 + 2 = 8.

–8 Both numbers are negative, so the sum is negative.

Add.

–5 + (–7)

Check It Out! Example 2a

When the signs are the same, find the sum of the absolute values.

Both numbers are negative, so the sum is negative.

–5 + (–7) = 5 + 7

5 + 7 = 12

–12

Check It Out! Example 2b

Add.

–13.5 + (–22.3)

When the signs are the same, find the sum of the absolute values.

–13.5 + (–22.3)

–35.8 Both numbers are negative so,the sum is negative.

13.5 + 22.3

Check It Out! Example 2c

Add.

x + (–68) for x = 52 First substitute 52 for x.

x + (–68) = 52 + (–68)

68 – 52

When the signs of the numbers are different, find the difference of the absolute values.

–16Use the sign of the number with the greater absolute value. The sum is negative.

Two numbers are opposites if their sum is 0. A number and its opposite are on opposite sides of zero on a number line, but are the same distance from zero. They have the same absolute value.

A number and its opposite are additive inverses.To subtract signed numbers, you can use additiveinverses.

11 – 6 = 5 11 + (–6) = 5

Additive inverses

Subtracting 6 is the sameas adding the inverse of 6.

Subtracting a number is the same as adding theopposite of the number.

Subtract.

–6.7 – 4.1

–6.7 – 4.1 = –6.7 + (–4.1) To subtract 4.1, add –4.1.

When the signs of the numbersare the same, find the sum of theabsolute values: 6.7 + 4.1 = 10.8.

= –10.8 Both numbers are negative, so the sum is negative.

Example 3A: Subtracting Real Numbers

Subtract.

5 – (–4)

5 − (–4) = 5 + 4

9

To subtract –4 add 4.

Find the sum of the absolute values.

Example 3B: Subtracting Real Numbers

Subtract.

Example 3C: Subtracting Real Numbers

First substitute for z.

To subtract , add .

Rewrite with a denominator of 10.

Example 3C Continued

Write the answer in the simplest form. Both numbers are negative, so the sum is negative.

When the signs of the numbers arethe same, find the sum of the absolute values: .

Subtract.

13 – 21

Check It Out! Example 3a

13 – 21 To subtract 21 add –21.

When the signs of the numbers are different, find the difference of the absolute values: 21 – 13 = 8.

Use the sign of the number with the greater absolute value.

–8

= 13 + (–21)

Check It Out! Example 3b

Subtract.

Both numbers are positive so, the sum is positive.

To subtract add .–3 12 3 1

2

When the signs of the numbers are the same, find the sum of the absolute values: = 4.3 1

212

+

4

x – (–12) for x = –14

Check It Out! Example 3c

Subtract.

x – (–12) = –14 – (–12) First substitute –14 for x.

–14 + (12) To subtract –12, add 12.

When the signs of the numbers are different, find the difference of the absolute values: 14 – 12 = 2.

Use the sign of the number with the greater absolute value.

–2

Example 4: Oceanography ApplicationAn iceberg extends 75 feet above the sea. The bottom of the iceberg is at an elevation of –247 feet. What is the height of the iceberg?Find the difference in the elevations of the top of the iceberg andthe bottom of the iceberg.

elevation at top of iceberg

75

Minus elevation at bottomof iceberg

–247

75 – (–247)

75 – (–247) = 75 + 247

= 322The height of the iceberg is 322 feet.

To subtract –247, add 247.Find the sum of the absolute values.

–

Check It Out! Example 4What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the oceans surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of –12,468 feet?

elevation at top of iceberg

550

Minus elevation of the Titanic –12,468

–

550 – (–12,468)550 – (–12,468) = 550 + 12,468

Distance from the iceberg to the Titanic is 13,018 feet.

To subtract –12,468, add 12,468.

Find the sum of the absolute values.= 13,018

Add or subtract using a number line.

1. –2 + 9 7 2. –5 – (–3) –2

Add or subtract.3. –23 + 42 19 4. 4.5 – (–3.7) 8.2

5.

Lesson Quiz

6. The temperature at 6:00 A.M. was –23°F.At 3:00 P.M. it was 18°F. Find the differencein the temperatures. 41°F

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